Still not sure if Play Store is the key element or Play Services. If it is the former you may have an option.
What would be desired in that case is to install the Nanodroid patched Play Store. Normally that would be a major pain and risk involving flashing some microG installer that includes the Play Store over what we already have (bad); or manually removing FakeStore, pushing Play Store to /system, manually setting up a permissions file and pushing that to /system, hoping the ROM boots, and hoping the ROM’s signature spoofing will apply to it.
Then I just found this…
[MODULE] fakestore2playstore - microG with License Verification + In-App-Purchases | XDA Forums
Replaces FakeStore with Play Store. Nothing more. Requirement: root - Magisk.
Further reading I see an /e/OS user in that thread. So far that seems like the most painless way to get what you need. Maybe.